Finite Group Databasebeta

← C2×#10624235 #13089431

Subgroups — C2×#10624235 #13089431

Order 1440.

998 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×D15C211yesmaximal, characteristic, complemented
2C22×(C3⋊D3)⋊D5C211yesmaximal, characteristic, complemented
3C2×C5×#638C211yesmaximal, characteristic, complemented
4C2×C3×#1220C211yesmaximal, characteristic, complemented
5#10624235C218yesmaximal, direct product factor
6C2×#67157C211yesmaximal, characteristic, complemented
7C2×#67156C211yesmaximal, characteristic, complemented
8C2×#67154C211yesmaximal, characteristic, complemented
9C2×#6790—31nomaximal
10C2×D5×#67—31nomaximal
11C23×C32×C5C2211yescharacteristic, complemented
12C22×C5×C3⋊D3C2211yescharacteristic, complemented
13C22×C3×D15—24no—
14C22×C3×D15C2211yescharacteristic
15C22×C3×D15C2214yescomplemented
16C22×C3×D15C2211yescharacteristic, complemented
17C22×C3×D15C2211yescharacteristic, complemented
18C2×(C3⋊D3)⋊D5—24no—
19C2×(C3⋊D3)⋊D5C2214yescomplemented
20C5×#638C2214yescomplemented
21C2×C5×C3⋊Dic3C2211yescharacteristic, complemented
22C3×#1220C2214yescomplemented
23C2×C3×Dic15C2211yescharacteristic, complemented
24#67157C2214yescomplemented
25#67156C2214yescomplemented
26#67154C2214yescomplemented
27C2×D3×#67—51nomaximal, Hall
28C23×D15D311yescharacteristic, complemented
29C23×C3×D5—31no—
30C22×D3×D5—31no—
31C22×D3×D5—31no—
32C2×#1220D311yescharacteristic, complemented
33C2×C5×#67—31no—
34C2×C5×#67—31no—
35C2×C5×#67—61no—
36C2×C3×C22⋊D5—31no—
37#6790—38no—
38D5×#67—38no—
39C2×#1203—31no—
40C2×#1202—31no—
41C2×#1202—31no—
42C2×#1201—31no—
43C2×#1200—31no—
44C2×D5×Dic3—31no—
45C22×C32×C5—22no—
46C22×C32×C5C2311yescharacteristic
47C22×C32×C5C2312yescomplemented
48C2×C5×C3⋊D3—22no—
49C2×C5×C3⋊D3C2312yescomplemented
50C2×C3×D15—22no—
51C2×C3×D15—22no—
52C2×C3×D15—28no—
53C2×C3×D15D414yescomplemented
54C2×C3×D15C2312yes—
55C2×C3×D15C2312yescomplemented
56C2×C3×D15C2312yescomplemented
57(C3⋊D3)⋊D5—28no—
58C5×C3⋊Dic3C2312yescomplemented
59C3×Dic15C2312yescomplemented
60C2×D4×D5—91noHall
61C23×C3×D3—51no—
62C22×D32—51no—
63C2×#638D511yescharacteristic, complemented
64C2×C3×#67—51no—
65D3×#67—58no—
66C2×#626—51no—
67C2×#625—51no—
68C2×D3×Dic3—51no—
69C23×C3×C5—21no—
70C23×C3×C5C2×D311yescharacteristic, complemented
71C23×C3×C5C2×D311yescharacteristic, complemented
72C22×D15—24no—
73C22×D15C2×D311yescharacteristic
74C22×D15C2×D314yescomplemented
75C22×D15C2×D311yescharacteristic, complemented
76C22×D15C2×D311yescharacteristic, complemented
77C22×C5×D3—31no—
78C22×C5×D3—31no—
79C22×C5×D3—61no—
80C22×C3×D5—31no—
81C22×C3×D5—31no—
82C22×C3×D5—31no—
83C22×C3×D5—34no—
84C22×C3×D5—64no—
85C2×D3×D5—34no—
86C2×D3×D5—34no—
87C2×D3×D5—64no—
88C2×D3×D5—64no—
89#1220C2×D314yescomplemented
90C2×Dic15C2×D311yescharacteristic, complemented
91C5×#67—34no—
92C5×#67—34no—
93C5×#67—64no—
94C2×C5×Dic3—31no—
95C2×C5×Dic3—31no—
96C2×C5×Dic3—61no—
97C3×C22⋊D5—34no—
98C2×C3×Dic5—31no—
99#1203—34no—
100#1202—34no—
101#1202—34no—
102#1201—34no—
103#1200—34no—
104D5×Dic3—34no—
105C22×#67—151no—
106C2×D3×D4—151no—
107C2×C32×C5—22no—
108C2×C32×C5C2×D412yescomplemented
109C2×C32×C5C2411yescharacteristic
110C5×C3⋊D3—22no—
111C3×D15—22no—
112C3×D15—22no—
113C3×D15C2×D414yescomplemented
114C23×D5—31no—
115C23×D5—91no—
116C2×C5×D4—91no—
117C2×C22⋊D5—31no—
118C2×C22⋊D5—91no—
119D4×D5—98no—
120C2×D20—91no—
121C2×C4×D5—91no—
122C23×C32C2×D511yescharacteristic, complemented
123C22×C3⋊D3C2×D511yescharacteristic, complemented
124C22×C3×D3—51no—
125C22×C3×D3—51no—
126C22×C3×D3—51no—
127C22×C3×D3—54no—
128C22×C3×D3—104no—
129C2×D32—54no—
130C2×D32—104no—
131#638C2×D514yescomplemented
132C2×C3⋊Dic3C2×D511yescharacteristic, complemented
133C3×#67—54no—
134C2×C3×Dic3—51no—
135#626—54no—
136#625—54no—
137D3×Dic3—54no—
138C22×C3×C5—22no—
139C22×C3×C5—22no—
140C22×C3×C5—21no—
141C22×C3×C5—22no—
142C22×C3×C5—42no—
143C22×C3×C5C22×D311yescharacteristic
144C22×C3×C5C22×D311yescharacteristic
145C22×C3×C5C22×D312yescomplemented
146C22×C3×C5C22×D312yescomplemented
147C2×D15—22no—
148C2×D15—22no—
149C2×D15—28no—
150C2×D15#6714yescomplemented
151C2×D15C22×D312yes—
152C2×D15C22×D312yescomplemented
153C2×D15C22×D312yescomplemented
154C2×C5×D3—32no—
155C2×C5×D3—32no—
156C2×C5×D3—62no—
157C2×C5×D3—62no—
158C2×C5×D3—62no—
159C2×C5×D3—122no—
160C2×C3×D5—34no—
161C2×C3×D5—32no—
162C2×C3×D5—32no—
163C2×C3×D5—32no—
164C2×C3×D5—68no—
165C2×C3×D5—62no—
166C2×C3×D5—62no—
167D3×D5—68no—
168D3×D5—68no—
169Dic15C22×D312yescomplemented
170C3×Dic5—32no—
171C5×Dic3—32no—
172C5×Dic3—32no—
173C5×Dic3—62no—
174C24×C3—151no—
175C23×D3—51no—
176C23×D3—151no—
177C23×D3—151no—
178C2×C3×D4—151no—
179C2×#67—31no—
180C2×#67—31no—
181C2×#67—51no—
182C2×#67—61no—
183C2×#67—151no—
184C2×#67—151no—
185C2×#67—151no—
186C2×#67—158no—
187C22×Dic3—151no—
188D3×D4—158no—
189C2×D12—151no—
190C2×C4×D3—151no—
191C32×C5C22×D411yescharacteristic, complemented, Hall
192C23×C5—91no—
193C23×C5D3211yescharacteristic, complemented
194C22×D5—31no—
195C22×D5—31no—
196C22×D5—31no—
197C22×D5—34no—
198C22×D5—64no—
199C22×D5—94no—
200C22×D5—184no—
201C5×D4—94no—
202C2×C4×C5—91no—
203C22⋊D5—34no—
204C22⋊D5—94no—
205C2×Dic5—31no—
206D20—94no—
207C4×D5—94no—
208C22×C32—22no—
209C22×C32C22×D511yescharacteristic
210C22×C32C22×D512yescomplemented
211C2×C3⋊D3—22no—
212C2×C3⋊D3C22×D512yescomplemented
213C2×C3×D3—54no—
214C2×C3×D3—52no—
215C2×C3×D3—52no—
216C2×C3×D3—52no—
217C2×C3×D3—108no—
218C2×C3×D3—102no—
219C2×C3×D3—102no—
220D32—108no—
221C3⋊Dic3C22×D512yescomplemented
222C3×Dic3—52no—
223C22×D4—451noSylow
224C2×C3×C5—22no—
225C2×C3×C5—22no—
226C2×C3×C5—21no—
227C2×C3×C5—22no—
228C2×C3×C5—42no—
229C2×C3×C5D3×D412yescomplemented
230C2×C3×C5C2×#6712yescomplemented
231C2×C3×C5C23×D311yescharacteristic
232C2×C3×C5C23×D311yescharacteristic
233D15—22no—
234D15—22no—
235D15C2×#6714yescomplemented
236C3×D5—34no—
237C3×D5—62no—
238C3×D5—62no—
239C5×D3—62no—
240C5×D3—62no—
241C5×D3—122no—
242C23×C3—21no—
243C23×C3—151no—
244C23×C3—151no—
245C23×C3—151no—
246C23×C3—154no—
247C23×C3—304no—
248C23×C3D3×D511yescharacteristic, complemented
249C23×C3D3×D511yescharacteristic, complemented
250C22×D3—31no—
251C22×D3—31no—
252C22×D3—51no—
253C22×D3—51no—
254C22×D3—51no—
255C22×D3—54no—
256C22×D3—61no—
257C22×D3—104no—
258C22×D3—154no—
259C22×D3—154no—
260C22×D3—304no—
261C22×D3—304no—
262C3×D4—154no—
263C2×C3×C4—151no—
264#67—34no—
265#67—34no—
266#67—54no—
267#67—64no—
268#67—154no—
269#67—154no—
270#67—154no—
271C2×Dic3—31no—
272C2×Dic3—31no—
273C2×Dic3—51no—
274C2×Dic3—61no—
275C2×Dic3—154no—
276D12—154no—
277C4×D3—154no—
278C22×C5—22no—
279C22×C5—92no—
280C22×C5—182no—
281C22×C5C2×D3211yescharacteristic
282C22×C5C2×D3212yescomplemented
283C2×D5—32no—
284C2×D5—34no—
285C2×D5—32no—
286C2×D5—32no—
287C2×D5—68no—
288C2×D5—62no—
289C2×D5—62no—
290C2×D5—188no—
291C4×C5—92no—
292Dic5—32no—
293C2×C32—22no—
294C2×C32D4×D512yescomplemented
295C2×C32C23×D511yescharacteristic
296C3⋊D3—22no—
297C3×D3—54no—
298C3×D3—102no—
299C3×D3—102no—
300C24—151no—
301C24—451no—
302C2×D4—91no—
303C2×D4—151no—
304C2×D4—451no—
305C2×D4—451no—
306C2×D4—458no—
307C22×C4—451no—
308C3×C5—21no—
309C3×C5C2×D3×D411yescharacteristic, complemented
310C3×C5C22×#6711yescharacteristic, complemented
311C22×C3—22no—
312C22×C3—22no—
313C22×C3—21no—
314C22×C3—22no—
315C22×C3—42no—
316C22×C3—154no—
317C22×C3—152no—
318C22×C3—152no—
319C22×C3—152no—
320C22×C3—308no—
321C22×C3—302no—
322C22×C3—302no—
323C22×C3C2×D3×D511yescharacteristic
324C22×C3C2×D3×D511yescharacteristic
325C22×C3C2×D3×D512yescomplemented
326C22×C3C2×D3×D512yescomplemented
327C2×D3—32no—
328C2×D3—32no—
329C2×D3—54no—
330C2×D3—52no—
331C2×D3—52no—
332C2×D3—52no—
333C2×D3—62no—
334C2×D3—62no—
335C2×D3—62no—
336C2×D3—108no—
337C2×D3—102no—
338C2×D3—102no—
339C2×D3—122no—
340C2×D3—308no—
341C2×D3—308no—
342C3×C4—152no—
343Dic3—32no—
344Dic3—32no—
345Dic3—52no—
346Dic3—62no—
347C2×C5—22no—
348C2×C5—182no—
349C2×C5D3×#6712yescomplemented
350C2×C5C22×D3211yescharacteristic
351D5—34no—
352D5—62no—
353D5—62no—
354C32C2×D4×D511yescharacteristic, complemented, Sylow
355C23—91no—
356C23—151no—
357C23—151no—
358C23—151no—
359C23—154no—
360C23—304no—
361C23—454no—
362C23—904no—
363C23(C3⋊D3)⋊D511yescharacteristic, complemented
364D4—94no—
365D4—154no—
366D4—454no—
367D4—454no—
368C2×C4—91no—
369C2×C4—151no—
370C2×C4—454no—
371C2×C3—22no—
372C2×C3—22no—
373C2×C3—21no—
374C2×C3—22no—
375C2×C3—42no—
376C2×C3—154no—
377C2×C3—302no—
378C2×C3—302no—
379C2×C3D5×#6712yescomplemented
380C2×C3#679012yescomplemented
381C2×C3C22×D3×D511yescharacteristic
382C2×C3C22×D3×D511yescharacteristic
383D3—54no—
384D3—62no—
385D3—62no—
386D3—102no—
387D3—102no—
388D3—122no—
389C5C2×D3×#6711yescharacteristic, complemented, Sylow
390C22—22no—
391C22—92no—
392C22—152no—
393C22—154no—
394C22—152no—
395C22—152no—
396C22—182no—
397C22—308no—
398C22—302no—
399C22—302no—
400C22—908no—
401C22C2×(C3⋊D3)⋊D511yescharacteristic
402C22C2×(C3⋊D3)⋊D512yescomplemented
403C4—92no—
404C4—152no—
405C3—21no—
406C3C2×D5×#6711yescharacteristic, complemented
407C3C2×#679011yescharacteristic, complemented
408C2—22no—
409C2—154no—
410C2—182no—
411C2—302no—
412C2—302no—
413C2#1062423512yesdirect product factor
414C2C22×(C3⋊D3)⋊D511yescharacteristic
subgroupG/H
centerC22C2×(C3⋊D3)⋊D5
commutator, cosocleC2×C32×C5C24
FrattiniC2C22×(C3⋊D3)⋊D5
FittingC23×C32×C5C22
socleC22×C32×C5C23
hypercenterC23(C3⋊D3)⋊D5
hypocenterC32×C5C22×D4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C23

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C23×C3×D15C2semidirect
2. C22×(C3⋊D3)⋊D5C2semidirect
3. C2×C5×#638C2semidirect
4. C2×C3×#1220C2semidirect
5. #10624235C2direct (N × H)
6. C2×#67157C2semidirect
7. C2×#67156C2semidirect
8. C2×#67154C2semidirect
11. C23×C32×C5C22semidirect
12. C22×C5×C3⋊D3C22semidirect
15. C22×C3×D15C22semidirect
16. C22×C3×D15C22semidirect
17. C22×C3×D15C22semidirect
19. C2×(C3⋊D3)⋊D5C22semidirect
20. C5×#638C22semidirect
21. C2×C5×C3⋊Dic3C22semidirect
22. C3×#1220C22semidirect
23. C2×C3×Dic15C22semidirect
24. #67157C22semidirect
25. #67156C22semidirect
26. #67154C22semidirect
28. C23×D15D3semidirect
32. C2×#1220D3semidirect
47. C22×C32×C5C23semidirect
49. C2×C5×C3⋊D3C23semidirect
53. C2×C3×D15D4semidirect
55. C2×C3×D15C23semidirect
56. C2×C3×D15C23semidirect
58. C5×C3⋊Dic3C23semidirect
59. C3×Dic15C23semidirect
63. C2×#638D5semidirect
70. C23×C3×C5C2×D3semidirect
71. C23×C3×C5C2×D3semidirect
74. C22×D15C2×D3semidirect
75. C22×D15C2×D3semidirect
76. C22×D15C2×D3semidirect
89. #1220C2×D3semidirect
90. C2×Dic15C2×D3semidirect
108. C2×C32×C5C2×D4semidirect
113. C3×D15C2×D4semidirect
122. C23×C32C2×D5semidirect
123. C22×C3⋊D3C2×D5semidirect
131. #638C2×D5semidirect
132. C2×C3⋊Dic3C2×D5semidirect
145. C22×C3×C5C22×D3semidirect
146. C22×C3×C5C22×D3semidirect
150. C2×D15#67semidirect
152. C2×D15C22×D3semidirect
153. C2×D15C22×D3semidirect
169. Dic15C22×D3semidirect
191. C32×C5C22×D4semidirect
193. C23×C5D32semidirect
210. C22×C32C22×D5semidirect
212. C2×C3⋊D3C22×D5semidirect
221. C3⋊Dic3C22×D5semidirect
229. C2×C3×C5D3×D4semidirect
230. C2×C3×C5C2×#67semidirect
235. D15C2×#67semidirect
248. C23×C3D3×D5semidirect
249. C23×C3D3×D5semidirect
282. C22×C5C2×D32semidirect
294. C2×C32D4×D5semidirect
309. C3×C5C2×D3×D4semidirect
310. C3×C5C22×#67semidirect
325. C22×C3C2×D3×D5semidirect
326. C22×C3C2×D3×D5semidirect
349. C2×C5D3×#67semidirect
354. C32C2×D4×D5semidirect
363. C23(C3⋊D3)⋊D5semidirect
379. C2×C3D5×#67semidirect
380. C2×C3#6790semidirect
389. C5C2×D3×#67semidirect
402. C22C2×(C3⋊D3)⋊D5semidirect
406. C3C2×D5×#67semidirect
407. C3C2×#6790semidirect
413. C2#10624235direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C23×C3×D152
C22×(C3⋊D3)⋊D52
C2×C5×#6382
C2×C3×#12202
#106242352
C2×#671572
C2×#671562
C2×#671542
C2×#67903
C2×D5×#673
C2×D3×#675